Math 2930 Spring 08 Prelim 1
1. Given the following differential equation
dy
dx
= 4
x

2
y,
y
(0) = 0
(1)
(a) Use Euler’s method to approximate the differential equation on the interval
x
= [0
,
2].
Use a step size of
h
= 1
/
2 and
clearly state
any equations used.
(b) Solve the differential equation
dy
dx
= 4
x

2
y
,
y
(0) = 0.
(c) What is the error is Euler’s method for
y
(2) in part (a)?
2. The population of a duck pond is governed by the following differential equation
dP
dt
= 2
P
(4

P
)
,
P
(0) = 2
(2)
(a) Solve for the population as a function of time,
P
(
t
).
(b) What is the limiting population?
3. Find the general solution to the differential equation
dy
dx
=
x
+ 2
y
y

2
x
(3)
For what initial conditions (if any) is the solution guaranteed to exist and be unique?
4. Given the differential equation below, answer the questions.
dy
dx
= (
y

1)(2

y
)(
y

r
)
(4)
(a) Calculate, then draw the bifurcation diagram for the system.
(b) Determine the stability of the critical points (equilibrium curves) in the bifurcation diagram.
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 Spring '07
 TERRELL,R
 Equations, 1 m, 2 m, 5 gal

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